COMPOSITIO MATHEMATICA

MICHAEL KERNER Abstract Compositio Mathematica, tome 4 (1937), p. 308-341

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Abstract differential geometry by Michael Kerner

1. Introduction. When building a differential geometry one starts usually with the conception of an "arithmetic" space of n whose properties are presumed to be known. The géométrie is then n-dimensional. Recently were also published some papers, as these of Michal, Kawaguchi, Conforto and others 1), which have replaced the arithmetic space by particular functional spaces. A farther step in this direction leads to the investigation of the differential geometry built upon an abstract space. To this problem is devoted the present paper. We transplant the géométrie research to a space of type (B) as . This space is also known as Banach space. It is a linear, normed and complete space 2) which embraces the n-dimensional space, the Hilbert space and many others as special cases. We attempt to build a géométrie (in general an affine) manifold on an underlying space of type (B) with the tendency to get a perfect conformity with differential geometry of n dimensions in the sense, that our theory must become identical with the former when the abstract space is taken in particular as n- dimensional. On account of this tendency one can consider our theory as a didactically new representation of the classical differential geometry which avoids the employing of coordinate and the flood of indices. The new representation yields of course also some inconveniences peculiar to it, and it were scarcely worth the trouble to introduce the new apparatus only for the sake of notation simplicity.

1) A detailed bibliography one can find in STRUIx’S Theory of Linear Connec- tions [Berlin 1934] and KAWAGUCHI’S paper [Journ. Hokkaido Imp. Univ. (1) 3 (1935), 103-106]. 2) Cf. BANACH, Théorie des opérations linéaires [warszawa 1932], 53. 309

In one regard only we decided in favour of a diversion from the classical theory. We defined the tangent spaces of a manifold and the vectors and in such a manner that they do not undergo any transformations. It is done without any deep- seated artifice and allows us to return easily in the case of n dimensions to the classical treatment. We wished to consider the tangent spaces as something absolute, and to submit to trans- formations only the representation of their connection. When passing to a more general ground one must renounce some notions and some properties of the examined object. The most important is that there are in an abstract géométrie manifold no tensors of contravariant order greater than one. It seems, however, that these play no natural role in geometry, and the absence of them corresponds somewhat to the essential circum- stances. The analogy between covariant and contravariant quantities is to be lost of course. In the general case we call the object of investigation the "manifold" and not the ,space,", because it is composed itself of spaces (namely the tangent spaces). Moreover, as to termino- logy, we follow the classical one though it is not well adapted to our purposes. We do it in order to facilitate the understanding of our notation which is not accepted in geometry. We shall use the word ,,function" in a general sense replacing by it the words "functional" and "operation". The géométrie word "represen- tation" will be also often employed in this aim.

Part I. Linear spaces.

2. Vectors in the affine linear space. We consider an abstract space of type (B), that is a linear, normed and complete space. We assume that the norm can be replaced by another one with the condition that both norms lead to the same determination of the limit. Hence it follows that both spaces (with two different allowable norms) are isomorphic 3) as spaces of type (B), the isomorphism being determined by the linear operation which associâtes every element with itself. The ratio of norms in both spaces is uniformly limited for all elements from above as from below.

3) Cf. BANACH, loc. cit., 180. 310

Such a space # with not completely defined norm we shall call an affine linear space. The capitals will denote the elements of the space F, and the small letters the real numbers. We call every point (element) A of the space # a contra- variant vector of this space. The notion of a contravariant vector new was a new is not (it also called vector),y but the term is introduced for distinction from a covariant vector. We especially that the notion of a contravariant vector is com- pletely independent of any coordinate transformations. We call every linear (additive and continuous) functional f(X} in the space f3 a covariant vector of this space. Therein X denotes an indefinite variable which can run all over the space Z - We do not deal here with the numerical value of the functional, but with the functional itself, as an operation. We shall often denote the covariant vectors also by the symbol f(.). The covariant vectors are elements of the space #f conjugate to &2 4 ). This space (that is the space of linear functionals ) is also normed, but, like in f3 , the norm is therein not completely determined. When given a covariant vector f(.) and a contravariant vector A, we call the numerical value f(A}, taken by the functional f(. ) for the element A, the inner () product of f(.) and A. If the inner product vanishes, the covariant vector f(. ) and the contravariant vector A are called orthogonal one to another. This notion is not new 5). If two spaces # and e"’ of type (B) are isomorphic, we shall call also the corresponding affine linear spaces isomorphic. Because of the allowed change of the norm in such spa- ces there is no notion of equivalence of spaces 6), the latter being not distinguished among the isomorphisms. As an example of isomorphic spaces one can consider a space, vectors of which are forces, and the ordinary configuration space. One can build inner products of covariant and contravariant vectors which belong to isomorphic spaces. One must only replace

4) Cf. BANACH, loc. cit., 188. s) Cf. BANACH, loc. cit., 59. 6) Cf. BANACH, loc. cit., 180. 311 the contravariant vector of one space by the corresponding vector of the other, the isomorphism being uniquely determined of course.

3. Tensors in the affine linear space. We use the word "" in a general sense without any symmetry restrictions. We define no contravariant tensors of order greater than one. This is the most important difference we meet with when dealing with the abstract geometry. in We call every n-linear functional f(Xi, X 2, ..., Xn ) the space 7ff a covariant tensor of order n of this space. We write also if the f(. , . .. , , . .), but this kind of notation is inconvenient, numerical value of n is not given. We remember that n-linear functionals are functionals linear relative to each their argument. A covariant tensor is said to be symmetric (resp. alternating), has the same if the n-linear functionalf(X1, X 2, ..., Xn) property. The inner (scalar ) product of a covariant tensor f(Xi, X 2, ... ,Xn) of order n and contravariant vectors Al, A 2, ..., is to be defined in like manner as for covariant vectors. If the number of vectors the the inner A 1, A2, ..., An equals to the order n of tensor, product f(A1’ A 2, ... , An ) is the numerical value of the mul- tilinear functional, when for all its arguments are substituted succesively the vectors Al, A 2, ..., An. The order of these vectors is to be respected. Only in the case of symmetric tensors it is indifferent. In general there are on the contrary n ! different products. If on the other hand the number m of vectors A l’ A 2, ... , A m is less than the order n of the tensor, the different inner products are covariant tensors of order (n -m ), for instance

In generalg there are productp formations, which become (nn--n>t ’n1 m) 11 identical, if the tensor is symmetric. We call every n-linear operation F(X1, X2, ..., Xn) which an associates with the elements Xl, X2, ..., Xn of the space 7ff element of the same space, a mixed tensor of contravariant order one and covariant order n of this space (also of total order n + 1). It

can be also written F(.,., ... , . ). The real numbers, the covariant and contravariant vectors, the covariant and mixed tensors are called all together quantities. A mixed tensor is said to be symmetric (resp. alternating), if same the n-linear operation F(X1, X2, ..., Xn ) has the property. 312

These terms refer only to the covariant behaviour of a mixed tensor. As to the inner product there are n ! different products by n contravariant vectors A 1, A 2, ... , An, as for instance F (A i, A 2, ..., A n) . These are themselves contravariant vectors (as elements of the n! E In the case of m vectors there are (n-m)! space ). (n-m) . products, as for instance F(A1, A2, ..., Am’ Xm+1’ ..., Xn), which are mixed tensors of covariant order (n -m ) (we shall for mixed tensors declare only the covariant order). All these products become identical, if the original mixed tensor is symmetric. In particular, the product of a mixed tensor F(.) of total order two by a contravariant vector A is a contravariant vector F(A ). We have yet to define the outer product. It is identical with the ordinary product, and may be defined only when at least one of two quantities is purely covariant. In other words, the outer product tzvo tensors of covariant f(Xi, X 2, ..., Xn ) and g(X1, X 2, ...,’ X m ) of order n and m Tespectively is the covariant tensor h(Xl, X2, ..., X.1m) = f(Xl, X2, ..., Xn) . g(Xn+1’ Xn+2’ ..., Xn+m) of order (n + m); the outer product of a covariant tensor tensor f(X1, X2, ..., Xn ) of order n by a mixed F(X1, X 2, ..., Xm) of covariant order m is the mixed tensor G(X1, X2, ..., Xn+m) === f(X1, X2, ..., Xn) . F(X.+15 Xn+2’...’ Xn+m) of covariant order (n+m). One defines likewise the outer product of a mixed tensor by a covariant one. The definitions embrace also the case of vectors, because the contravariant vector is a special case of mixed tensor (of covariant order zero), and the covariant vector is a special case of covariant tensor (of order one). The outer product is not permutable.

4. Metric linear space. We introduce in an affine linear space a quadratic metric by defining a covariant fundamental tensor g(X, Y) of order two. We assume that 1. the tensor g(X, Y) is symmetric,

2. there is such a positive number l that for every X (the metric is "uniformly" positive definite). 313

The second condition is independent on eventual allowable change of norm. An affine linear space with a metric which satisfies both enumerated conditions is called a metric linear space. The following inequality is true

where m is an appropriate positive number 1). Hence we obtain in particular the inequality. analogous to (4.2). If -97 is a metric linear space, every contra variant vector A has an absolute value equal to ilg(A, A). The inequalities (4.2) and (4.4) allow us to prove that the square g(X, X) of the absolute value of X is an allowable norm. We omit this easy demonstration. In a metric linear space R to each contravariant vector A corresponds a covariant vector

which is to be denoted by the same letter as the former (but small, of course). We call a (X) the covariant vector conjugate to A. It is an element of the space .5fl conjugate to R , that is, of the space of linear functionals in 572: . However the corres- pondence

is in general not reversible for every a(X). It carries the whole space £3i into a linear set 3- of the conjugate space R . The operation (4.6), where a(X) is defined by (4.5), admits a uniquely determined inverse operation in Y, and both opera- tions are linear. Indeed, if there were two contravariant vectors A and B, for which

it would follow for every X

’1) See KERNER, Die Differentiale in der allgemeinen Analysis [Ann. of Math. (2) 34 (1933), 548]. 314 and in particular

contradicting to (4.2 ), when A and B are not equal. The inversion of (4.6) is thus uniquely determined. As to linearity, both opera- tions are obviously additive. There remains the continuity proof. It follows from (4.3) that

and hence, according to the definition of the functional’s norm,

The direct operation is thereby continuous. On the other hand, it follows from (4.2) that and hence

Thus, the inverse operation is continuous, too. It follows from this continuity that the linear set T of the space R is closed. We wish to observe that 3 is total 8). For, if A is such an element that

for every functional f ( X ) of the set .!7, we have in particular

or whence, according to (4.2), it follows

The most important is the case when the set 5 coincides with the whole space R . In other words, the correspondence (4.6) is then reversible for every a(X). To each covariant vector a(X) there corresponds a contravariant vector A, for which

8) Cf. BANACH, loc. cit., 42. 315

In such a case we call the metric spanning (because the conjugate covariant vectors relative to this metric span the conjugate space R ). If the metric is spanning, the space !7l is isomorphic with the conjugate space j3!f . Since this property is attributed to spaces only exceptionally, one cannot in general introduce a spanning metric into a space. We shall soon see that not only a spanning metric, but also an ordinary (positive definite) metric cannot be introduced in general. We turn now to the special case of separable space. According to what was said above we can take the square g(X, X) of the absolute value as norm of X. Then it is easy to see that for our space ÉÙ are satisfied all the axioms which characterize the real Hilbert space, when regarding the bilinear functional g(X, Y) as the inner product in this space 9). This conception of Hilbert space embraces also the case of the euclidean space of a finite number of dimensions. Hence, if the original space of type (B) is separable, it must be isomorphic to Hilbert space. Thus, not into every space can one introduce a quadratic positive definite metric, if no restrictions about spanning are made. We return to the separable metric linear space R . If normed by g(X, X) it is a Hilbert space. In such a space every linear functional can be represented by g(X, A ), where, as said, g(X, A ) denotes the inner product of X by A, and A is an appropriate element of the space R . Therefore every covariant vector is conjugated to a contra variant one. We have to note an important theorem: In a separable space each quadratic positive definite metric is spanning.

5. l’ectors and tensors in the metric linear space. In a metric linear space we can define the inner product of two contravariant vectors A and B. It is equal to the inner product of the covariant vector a(X) conjugate to A by the second contravariant vector B, that is, to a(B). The inner product is permutable, since

9) See NEUMAAN, Mathematische Grundlagen der Quantenmechanik [Berlin 1932], 19-24. The axioms are built for complex Hilbert space. 316

In a separable space it is identical with the inner product in the isomorphic Hilbert space. If the metric is spanning, the inner product of two covariant vectors a(X) and b(X) can be defined. It is equal to the inner product of one vector a(X) by the contravariant vector B, to which b(X) is conjugate, that is, to a(B). It is permutable, too, because of the same equalities (5.1). When the metric is not spanning, the last definition is of value only if at least one of the covariant vectors a(X) and b(X) belongs to the set T of the conjugate space .9l . If the inner product of two covariant or contravariant vectors vanishes, these are called orthogonal one to another. To each mixed tensor A (Y, Z, ... ) in the space L91 corres- ponds a conjugated covariant tensor g[X, A ( Y, Z, ...)] of the same total order (the process leading from the former to the latter is analogous to the usual lowering of a contravariant index). If the metric is spanning, the inverse process to the former can be defined. Let a(X, Y, Z, ...) to be a covariant tensor. We wish, for instance, to "bring up the first index" which corres- ponds to the variable X. We must for this purpose determine a mixed tensor A (Y, Z, ... ), for which

We have to solve for fixed Y, Z,... the equation

relative to U. This equation has a unique solution U which depends on Y, Z,... We ought to prove that this solution is a multilinear operation (mixed tensor). The additiveness of this operation is obvious on account of the unicity of solution of (5.3). We yet must still prove that U is a continuous function

of Y for given Z, ... The proof for the remaining variables is

the same. Let the values of Z, ... to be fixed. We have then the inequality.

for an appropriate positive number k. Hence, on account of (5.3),

We put here 317 and conclude from (4.2)

or whence the continuity relative to Y follows. Conclusively, the solution of (5.3) is a mixed tensor A (Y, Z, ... ) which satis- fies (5.2). When wishing to "bring up the second index", which corres- ponds to Y, one must determine a mixed tensor A (X, Z, ... ) from the equation

If the original covariant tensor a(X, Y, Z, ... ) is symmetric, all mixed tensors obtained in such a manner are identical. The notion of inner product of tensors can be also generalized. In a metric linear space with a spanning metric one can for instance obtain the inner product of a covariant tensor a(X, Y) by covariant vectors b (X ) and c(X), when forming the inner product a(B, C) of a(X, Y) by the contravariant vectors B and C, to which b(X) and c(X ) are conjugate. It is not worth the trouble to enumerate all the possible cases on this account.

Part II. Général .

6. Bare manifolds. Let I be a space of type (B). We call it a substratum (underlying space). We suppose that to ev ery point P of the substratum 9 or of a region !fJ of the space 57 there corresponds an affine linear space !7; which is isomorphic to the substratum as a space of type (B). This space .3; we call the tangent space at point P. The tangent spaces at different points of gare therefore isomorphic to each other. The totality of tangent spaces under their connection with the substratum, which will be stated beneath, is called a bare (geometric) manifold and denoted by X. Because of the isomorphism of all tangent spaces Tp we can represent them all on an arbitrarily chosen tangent space. There are of course many ways of such a representation. We choose fixed linear transformations of all tangent spaces into one 318 of them, and decide to denote by the same letter, say X, the elements of different spaces, associated one to another by these transformations. The tangent spaces being affine linear, we replace the norms therein by the norm in the distinguished tangent space. In such a manner all the tangent spaces Tp- become identical, and we can speak of a unique tangent space Y. The way of transforming all the spaces into one of them does not play any geometric role and can be replaced by another one. But for continuity and differentiation considerations one must choose a fixed transformation and conserve it during all the investigations. Now we formulate two conditions defining bare manifolds : the connection between the tangent spaces and the substratum, and the allowable substratum changes. 1. The condition of connection between the tangent spaces and the substratum. For every point P of the substratum I (resp. of a region B there is given a one to one linear transfor- mation of the space .57 into the whole tangent space 5 (or p ). This transformation and its inverse are continuous func- tions of the point P. We denote the direct linear transformation (function) by

liere Q is an element of the space iI, and X one of the space y. The dependence on the point P is marked by an index. The inverse function is also linear 1°) and will be denoted by

Both functions (6.1) and (6.2) are continuous with respect to P. It is interesting that the continuity of (6.1) does not imply that of (6.2) 11).

10) See BANACH, loc. cit., 41. 11 ) ii"e wish to show it on an example. Let P, Q, X to be elements of the Hilbert space formed by sequences: pi, p2, ... ; ql, q2, · · .; x1, x2, ... ; let us denote by

the greatest integer contained in . Then let us define the function ( 6.1 ) as follows :

when P t 0:

when P = 0:

One can prove that the function is continuous in respect to P for P = 0, while

the inverse function is not (it is easy to see the last fact when putting Xi = . i1 ). The discontinuities on the spheres IBP il= n (n integer) can be easily smoothed. 319

If the preliminary transformation of all tangent spaces into one of them were changed, then the function (6.1) would alter. But now this change should be restricted to be continuous with respect to P. Otherwise the altered function Fp(Q) would cease to be continuous. In order to understand better the first condition one can imagine the elements Q as ,infinitésimal displacements" at the point of the substratum, and the elements X as something similar in the tangent space. We reject the usual identification of the neighbourhood of a point in the substratum and in the tangent space. 2. The condition of allowable substratum changes. It is allowed to transform the region B of the spaces I one to one into a region B’ of another space I’ of type (B) isomorphie to I, by means of a function

This function and the inverse function must be continuous and must have the continuous first differentials

The space I’ is then a new substratum under the condition that the function X = Fp(Q) will be replaced by

When changing the substratum 9, we consider the point P and the corresponding point P’ of 1’ as the same point of the manifold. One can say that the function Fp(Q ) is transformed by the linear function tangent to Pp,. The tangent space stays invariable! The substratum change induces only the mutual correspondence between the substratum and the tangent space. The situation is the same as in the case of parametric representation of a curve. The substratum plays the role of a parameter. The change of substratum corresponds with the change of the parameter. We shall not have to deal at all with transformation of tangent spaces. The changes of substratum form of course a group of trans- formations. 320

The totality of all tangent spaces with their connection with the substratum (condition 1 ) under admission of allowable substratum changes (condition 2) is called a bare (geometric ) manifold of class one. The manifold X is of class two, if the functions Fp(Q) and Op(X) have the continuotis first differentials and the allowable functions Pp, and Pp the continuous second differentials

and so on. The differentials (6.5) are both symmetric 12). It follows from the identity

by differentiating

The formula (6.4) implies

and

whence

or, on account of (6.6),

If the manifold X is of class two, we can differentiate the identity

getting thus

or, by substituting

12) See KERNER, Ioe. cit. in footnote 7, 549. 321

We denote by Tp(X, Y) the function obtained from (6.9) by alternating, that is

One can prove that the symmetry of (6.9) [or the vanishing of Tp(X, Y ) ] is a necessary and sufficient condition that the tangent space (already transformed, of course) can be chosen in some sense itself as a particular substratum. The correspondence (6.1) becomes than a simple translation. We omit these con- siderations, because they have no significance for us, the pre- liminary transformation of tangent spaces into one of them [which influences the function Fp(Q)] having been taken accidentally.

7. Vectors -and tensors in the bare manifold. If for each point P of a region ..9J of the substratum / is defined a geometric quantity in the tangent space, and if this quantity is a continuous function of the point P, we speak of a geometric quantity in the manifold X. Properly spoken, we have to deal with a field of quantities. Thus we can consider in the manifold X scalars (that is real numbers depending on the point P), contravariant and covariant vectors, covariant and mixed tensors. The geometric quantities are quantities in tangent spaces, which do not depend on the substratum choice. Only when the definition of a function of proper type is bound to the substratum, one must prove the independence of this function on the substratum in order to know it to be a quantity. The quantities shall be denoted by ap (scalars), AP (contra- variant vectors), ap (X ) (covariant vectors), ap(X, Y), Ap(X), and so on, the dependence on the point P being brought into evidence by a subscript. But we shall often omit this subscript. When the symbols with a "prime" are attached to the changed substratum I’, the following conditions must hold for quantities

and so on.

A process which leads from geometric quantities to other ones is called invariantive. To each quantity corresponds in every substratum a function 322 of P which arises by representing the quantity on the substratum by means of the transformation Fp(Q), resp. f/>p(X). Thus, there corresponds to a contravariant vector A p the element OEp (A p ), to a covariant vector ap(X)the linear functional ap [Fp (Q)], to a mixed tensor Ap(X) the linear operation f/>p{Ap[Fp(Q)J}, and so on. We shall denote these functions by dashed letters, as A p, îip(Q), A p( Q). They depend on the substratum choice and coincide in a n-dimensional manifold with the quantities in the usual sense. In the last paragraph we have introduced for a manifold of class two the function T p(X, Y), the definition of which has been attached to the substratum. Nevertheless, we shall now prove that it is a mixed tensor of covariant order two. We differentiate the identity (6.4) and alternate, having regard to the symmetry of the second differential d2Pp, (Q’, R’ ),

Now we put here

and obtain with respect to (6.7), according to the definition (6.10),

whence it follows that Tp(X, Y) is a tensor.

8. Tangent differentials. We shall introduce in a manifold of class two a process which will be called tangent differentiation. According to what was mentioned above each quantity can be represented on the substratum by a function which is obtained by transforming this quantity by means of FP(Q ) or 0,(X). We form the differential of this function, and transform it backwards to the tangent space. The function obtained in such a manner we call the tangent differential and denote by the sign o. In some sense we have a formal relation

We shall explain it better by discussing several special cases. 323

Let fp be a scalar which has the continuous first differential dfp(Q). It is not necessary to transform it to the substratum. Then, we must differentiate it, getting dfp(Q), and transform this to the tangent space, obtaining

We wish to prove that this tangent differential of a scalar is a covariant tensor. Indeed, when effecting the change Pp, of substratum, we put whence it follows by differentiating that

Putting we obtain or, by virtue of that is

In such a manner, tangent differentiating of a scalar is an invariantive process. The covariant vector (8.1) is called the gradient of the scalar fp. Now let Ap be a contravariant vector which has the continuous first differential dA p(Q ). To get its tangent differential, we trans- form it into tPp(Ap), belonging to the substratum I, then differentiate and at last represent the obtained function backwards by

The first term of the right member does not depend on the substratum; but the second term, involving the operation dq)p(Q; R), does. Thus, the tangent differentiating of a contra- variant vector is not invariantive; no more is it so in general. We alternate (8.2), after having put X = Bp, and respecting (6.10), 324

The right member does not depend on the substratum, and we see that the left member is a contra variant tensor. Now we pass to the covariant vector ap(X) which has the continuous first differential dap(X; Q). The first transformation leads to ap[Fp(Q)], the differentiation to and then the inverse transformation yields

Also here the last term destroys the invariance. By alternating (8.4), having respect to (6.10), we get

whence it follows that the left member is a covariant alternating tensor of order two. This tensor is called the rotation of the co- variant vector ap(X). These examples show how to form the tangent differential of quantities. It is in some sense a representation of the differentia- tion in the substratum on the tangent space. The tangent differen- tials obey the same formal rules as the ordinary differential. If ap(X) is itself a tangent differential of a scalar fp, it follows from the way of forming the tangent differential that

Thus, if the scalar fp has the continuous second differential d2fp(Q, R), its second tangent differential -b2fp(X, Y) is symmetric. It is evident that the symmetry of oap(X; Y) is also a sufficient condition that ap(X) is a gradient. The analogous consideration for a contravariant vector Ap shows that 325 whence it follows that Z2Ap(X, Y) is symmetric, if A p has the continuous second differential.

9. . We turn now to the principal object of the present paper. We shall define in a bare manifold X of class two the covariant differential. This determines in the manifold the affine connection. A manifold with an affine connection is called an affine manifold and is denoted by -Z. We define the covariant differential by the following five assumptions: 1. The covariant differential of a quantity is a quantity whose covariant order is one greater than that of the given quantity. 2. The covariant differential of a scalar is equal to its tangent differential. 3. The covariant differential is a linear function of the quantity and its tangent differential, taken together. 4. The covariant differential of the outer product of two quan- tities obeys the same rule as the ordinary differential. 5. The covariant differential of the inner product of tzvo quan- tities obeys the same rule as the ordinary differential 13) . We shall denote the covariant differential by ô. The covariant differentials of a, A, a(X), A(X) are according to 1. ôa(X), ôA (X), ôa (X ; Y), ôA (X; Y) (we omit from now the subscript P). By virtue of 2. we have for a scalar f

We pass now to the covariant differential bA (X) of a contra- variant vector A. The assumption 3. means that ôA (X) is a linear function W of the pair [A, ZA (X)] ,

When putting here

we get

13) One can make assumptions involving much less than ours, but it were too long to derive the constitution of the covariant differential from them. 326

This equality is not quite explicit, since the function 1JI can depend also on the element X. By changing of notation and putting this dependence in evidence we can write where A and jT are linear functions of their first arguments. Now we must apply the last formula to the contravariant vector equal to the outer product f A, having regard to the assumption 4. and to (9.1),

whence for every A. In other words, A denotes the function independent of X and equal to the first argument A. Thus (9.2) becomes

From the assumption 1. it follows that bA (X) must be a linear function of X, so that the covariant differential of the contra- variant vector A is given by (9.3), where r(A, X ) is a bilinear operation of both arguments A and X. Every linear operation T(X, Y) [exactly -P,(X, Y)], which as- sociates with X, Y, belonging to the tangent space, an element of the same space, de termines an affine connection and can be called also the affine connection. We shall see that it is not a mixed tensor. That T(X, Y ) can be taken arbitrary, follows from the fact that the covariant differential (9.3) and those determined beneath satisfy the assumptions 1-5. It is supposed of course tha,t, when changing the substratum, T(X, Y ) is transformed in such manner that the covariant differential rests unaltered (is a quantity). The next step is to determine the covariant differential ba(X; Y ) of a covariant vector a(X). We choose an arbitrary contra variant vector A and form the inner product a(A ). The covariant differen- tial of this product (which is a scalar) is equal to

We apply on the other hand to a(A ) the rule 5. for obtaining the same differential 327

When comparing both expressions, we get with regard to (9.3)

The vector A being taken arbitrary, it follows from this equality that

that is the rule of covariant differentiating a covariant vector. To obtain the covariant differential ôA (X ; Y ) of a mixed tensor A (X), we form the inner product A (B) by an arbitrary contra- variant vector B and apply to it (9.3)

and on the other hand the rule 5.

The comparison of both expressions yields the covariant differen- tial of a mixed tensor of total order two

As last example we determine the covariant differential ôa(X, Y ; Z) of a covariant tensor a(X, Y). We form the inner product a(A, Y), which is a covariant vector, and apply to it (9.4)

and on the other hand by the rule 5.

The comparison of both expressions gives the covariant differen- tial of the covariant tensor of order two

The structure of the formulae (9.1), (9.3), (9.4), (9.5), (9.6) shows how to proceed in the general case of a tensor. It is for instance for a mixed tensor A(X, Y) of covariant order two 328

10. Torsion. Let us consider the formula (9.3), that is

According to the assumption 1. of the last paragraph ôA (X) is a mixed tensor, that is a function which does not depend on the substratum choice. Since ôA (X), determined by (8.2), has not the same property, the operation F(X, Y) is not a tensor; it depends on the substratum choice. The process ô is invariantive, while the process determined by F(. , X) is not. It follows from (8.2) that

does not depend on the substratum. Hence one can derive the rule of transformation of h(X, Y). We omit this matter. The invariance of (10.1) implies that of the alternated form [see the definition (6.10) of T(X, Y)]

Since T(X, Y) is a tensor, we conclude that

is a tensor, too. This alternating mixed tensor of covariant order two we call the torsion of the affine manifold. In case the torsion S(X, Y) vanishes, that is, of

we call the affine manifold symmetric and denote by A. If again

where s(X) is a covariant vector, we call the manifold semi- symmetric. According to (9.4) the rotation (8.5) of a covariant vector a(X ) is equal in a general affine manifold to

in a semisymmetric manifold to

and in a symmetric manifold to 329

11. Parallel displacement. Let g to be an affine manifold of class two. To the covariant differential bA (X) there corresponds in every substratum the function where, according to paragraph 8,

Let OE to be a curve of class one, represented by the function which admits the continuous first derivative P. If P(t), dpdt - the contravariant vector has the continuous first differential, we set Q = P in (11.1) and form the corresponding function in the tangent space

We call this expression the of A along the curve Ë. It is equal to

One can call the denote it and F p( P)) tangentg derivative, byy dt1 prove that it is a contravariant vector. The same reasonning yields the covariant derivative of a covariant vector

All properties of the covariant differential, such as the rule of differentiating outer and inner products, can be transported to the covariant derivative. If, in particular, the covariant derivative of a quantity vanishes, this quantity is said to be suhmitted to a parallel displacement along the curve. According to (11.3) the equation determining the parallel displacement of a contravariant vector is

or, if we pass to the substratum, 330

This equation can be written, when putting in the form

That is a linear differential equation relative to the function A (t) 14). It is equivalent to the equation

relative to the vector A ( t ) itself. If the value A of A (t ) for a given t, say t = 0, is determined, there is just one solution of (11.6), equal to A for t = 0. For a fixed t this solution depends linearly on the initial value A. Since the right member of (11.5) or (11.6) is linear relative to P, this equation is invariant under some change of the variable t. The solution A (t) at a determined point P of the manifold depends therefore only on the curve Ë, but not on its parametric representation, and rests also unaltered, when changing the sense of the curve. Thus, if Pl and P2 are two points of the curve 0152, to each vector at one point is associated a vector at the other, and vice-versa. This is a linear one to one correspondence of two tangent spaces at Pl and at P2. When transported parallelly the tangent space undergoes a linear transformation which is a con- tinuous function of the parameter t. The relation (11.4) leads to the following differential equation, determining the parallel displacement of a covariant vector a(X) [or a(X ; t), when bringing in evidence its dependence on t]

which is of more complicated type. We can avoid the discussion of this equation, when reducing the investigation of a covariant vector to that of a contravariant one. This can be done by virtue of the fact that the inner product of a(X ) by an arbitrary contra- variant vector A is invariant under parallel displacement. In other

14) As to the theory of differential equation in abstract spaces, cf. KERNER, Gewöhnliche Differentialgleichungen der allgemeinen Analysis [Prace mat. fiz. 40 (1932), 47201367]. 331 words, the covariant vectors undergo, when displaced parallelly, a transformation which is conjugate to that of contravariant vectors. The same reasonning allows to determine parallel displacement of other quantities.

12. Pseudometric. For the sake of some specializations it is convenient to introduce, if possible, a covariant of order two, g(X, Y) or gp(X, Y), which satisfies all assumptions of paragraph 4. Since this tensor does not play in all cases the role of metric, we shall call it pseudometric. We suppose in particular it to be spanning. We denote by q(X, Y, Z) the covariant differential of the pseudometric

or

We effect therein two permutations of X, Y, Z.

We add the second and the third equation and subtract from the result the first one, having regard to the definition (10.2) of the torsion and to the symmetry of g(X, Y),

If we denote the left member, which depends on g, q, S, in short by u(X, Y, Z), we get

Since the pseudometric g(X, Y) is spanning, the equation (12.3) determines in a unique way the function r(X, Y) for each X, Y. Its linearity is obvious. The affine connectio-n is therefore uniquely defined by means of the pseudometric g(X, Y), its covariant differential q(X, Y, Z) and the torsion S(X, Y). The first term (12.2) may be called the three-index symbol of 332

Christoffel of first kind, and will be denoted in short by c(X, Y, Z). The unique solution relative to C(X, Y) of the equation g[z, C(X, Y)] = c(X, Y, Z) may be called the three-index symbol of Christoffel of second kind. If the pseudometric g(X, Y) is not spanning, the resolution of (12.2) is possible only for certain determinations of the tensor q(X, Y, Z). Namely for these ones, for which the left member of (12.2), as functional of Z, belongs for each X, Y to the part y- of the space conjugate to the tangent space, according to the results of paragraph 4.

13. Different specializations of the affine manifold. If, in parti- cular,

where q(Z) is a covariant vector, we get the conf ormal connection. It follows for such a manifold from (13.1) when putting that

Therefore, when transporting parallelly two contravariant vectors along a curve, the ratio of their absolute values (defined by means of the pseudometric) rests invariant. Also their angle, defined by

is unchanged. A symmetric and conformal connection is called a Weyl con- nection. The equation (12.2) takes in this case the form

When denoting by Q(X, Y) the solution of the equation it follows from (13.2) 333

If in the place of (13.1) we assume that the connection is called metric. From (13.3) it follows that, when transporting parallelly a contravariant vector along a curve, its absolute value is constant. In a metric manifold the pseudometric g(X, Y) is called the metric of the manifold. Each tangent space of such a manifold is metric linear. A symmetric and metric connection is called a riernannian connection. We have in this case

The function T(X, Y) is identical with the three-index symbol of the second kind. For a curve OE of class one represented by P(t) [t1 tt2] in a riemannian manifold the length is defined by the integral

or

It is obviously a scalar (does not depend on the substratum choice). An example of riemannian manifold with a Hilbert’s substratum is given by the so-called "Riemann-Hilbert space" 15).

14. . Let 0152 to be a curve of class two represented by P(t) in an affine manifold L of class three. To P corresponds in the tangent space Tp the contravariant vector

This vector determines in the tangent space the direction of the curve OE. If the covariant derivative of Z(t) along the curve @ has in each point the direction of Z(t) itself, the curve OE is called a of the manifold L . To find geodesics one must solve the equation

15) See KERNER, Extremum dans l’espace hilbertien [Annali di mat. (4) 10 (1932), 1198-202]. 334

or, according to (11.3), where k (t ) is an arbitrary continuous numerical function of t. The fact that Z(t) is defined only along OE can be easily removed. We shall transform the equation (14.2), that is to the substratum where F(Q, Q ) is defined by (11.2). The first term is equal to the second derivative P of P(t), and the equation takes the form

Let Po to be a fixed point of / and A a vector at this point. Then Op(A) is the corresponding element of the substratum. According to the general theory of differential equations 14) the equation (14.3) has (we assume for an affine manifold of class three that T(X, Y) has a continuous first differential) just one solution P(t) in the neighbourhood of Po, for which

Thus, in each direction through a point Po there is a unique geodesic. If we put

by using the new parameter u we reduce the equation (14.3) to

and

The geodesics therefore do not depend on the arbitrar function k( t) . Because of the structure of their equation they depend only 335 on the symmetric part of T(X, Y ) and do not alter when changing the torsion S(X, Y). In the case of a riemannian connection (14.5) takes the form

which can be also written

or, according to the definition of c(X, Y, Z),

We transform this equation to the substratum

where Q is an arbitrary element. This can be also derived from (14.4) by means of transforming to I the definition equations of C(X, Y) and c(X, Y, Z). We shall prove that the geodesics of a riemannian manifold are extremals of the integral (13.5) or (13.4), which defines the length. The theory of extremals is till now developed only in the special case of Hilbert space 16), but it is easy to generalize it to arbitrary spaces of type ( B ) by means of the integral notion. If we put the integral (13.4) becomes

and its extremals are defined by the Euler equation where Q is arbitrary. We specialize the parameter t by the condition

Then we have

16) See KER1B4’"ER, Ioc. cit. in footnote 15, 183-202, 336

The Euler equation (14.7) takes the form

or equivalent to (14.6).

15. Curvature tensor. We suppose the affine manifold L to be of class three and the contravariant vector A to have a con- tinuous second differential. We form the iteration ô2A (X, Y) and alternate it, getting the tensor

From it follows, according to (9.5), that

The term ()2A(X, Y) is symmetric by virtue of (8.6). By alter- nating we get

We introduce the notation

The formula (15.1) becomes

The function R(X, Y, Z) is therefore a mixed tensor of covariant order two alternating in regard to both last variables Y, Z. We call it the curvature tensor of the affine manifold L . All the identities which are usually derived for the curvature tensor can be proved also in our general case. 337

16. Teleparallelism. If two points Po and Pi are joined by two curves @i and 01522 the parallel displacement of a vector A at Po along both curves leads in general to different vectors B1 and B2 at P. If,, on the contrary, the vectors at P do not depend on the joining curve 0152, at least in a neighbourhood of Po, we say that in the affine. manifold L there is a teleparallelism. We wish to find necessary and sufficient conditions for teleparallelism. Let A o to be an arbitrary contravariant vector at Po, the manifold if being of class three. The necessary and sufficient condition for teleparallelism is, that for each A o there exists in the neighbourhood of Po a vector A, for which the right member of (11.3) vanishes along each curve. This is equivalent to the condition

for each X. Thûs, the integrability of (16.1) is the necessary and sufficient condition for teleparallelism. From (16.1) it follows by tangent differentiating that

or

On account of (16.1) and by alternating we get a tensorial relation

This necessary condition leads by virtue of (15.3) to

and since the point Po and the value Ao were arbitrary, to

Thus, the vanishing of the curvature tensor is a necessary condition for teleparallelism. Before we prove that it is also sufficient, we must formulate the following theorem about the partial differential equations in the space of type (B): Let Y[H, As E] be a function which associates with each II 338 in neighbourhood of IIo in H-space, each A in neighbourhood of Ilo in A-space and each E of H-space an element of A-space. Let Y have continuous second differentials relative to II and A, and let it be linear relative to E. Then a necessary and sufficient condition that the equation relative to the unknown function A(II), have in some neighbour- hood of IIo just one solution Il (II ), for which consists in the symmetry of relative to E, H. We do not prove this theorem, since it can be done in a like manner as for the theory of differentials 17). It seems that even the assumption about the existence of the second differential of "tJf(II, A, E) is superfluous. Now we turn to the proof that (16.2) is a sufficient condition for teleparallelism. On account of what was assumed about the second differential in the last theorem we must suppose the affine manifold -9"to be of class four, but this could probably be avoided. The equation (16.1), that is

is equivalent to the equation

in the substratum. But (16.6) is of the form (16.3), where the point P is to be put in the place of II. The expression (16.5) becomes

to which corresponds in the tangent space the function

And the symmetry of (16.5) is equiiralent to that of (16.7),

17) Cf. KERNER, loc. cit. in footnote 7, 5462013557. 339 which is on its part equivalent to the vanishing of R (X, Y, Z). Thus is proved again the necessity and also the sufficiency of the condition (16.2).

17. Euclidean manifolds. We call an affine manifold g affine euclidean, if there exists such a substratum I0 that, when transporting parallelly a contravariant vector A, the corresponding element A = q)p(A) of this substratum is constant. In an affine euclidean manifold we have therefore for the substratum I0

and for every substratum I(because the torsion is a tensor)

There exists obviously in such a manifold teleparallelism, and in virtue of what was said in the last paragraph we have

We wish to prove that the necessary conditions (17.1) and (17.2) for an affine euclidean manifold are also sufficient. Indeed, if (17.2) is satisfied, we have teleparallelism, and the equation (16.6) has just one solution A p, for which in a fixed point Po

R being an arbitrary element of the substratum I. We denote A p by A p ( R ), bringing in evidence its dependency on R. According to what was said in paragraph 11 about parallel displacement Ap(R) is a linear function of R and has a uniquely determined inverse relative to R. After having defined Ap(R), we form the equation which is also of type (16.3). The unknown function Pp. associates with P’ belonging to 57 points of the same space. The condition of integrability (16.5) takes here the form

or, according to (16.6), 340

This equality is satisfied by virtue of (17.1). The equation (17.3) is therefore integrable. The function Pp,, which is defined by (17.3) with the initial condition

can be taken to be a substratum change. It transforms the substratum ,-CÇ7 into a new substratum I’ which is formed by the same abstract space, but with another correspondence to tangent spaces. To show that PP. is a substratum change we ought to determine also its inverse Pp. This latter can be defined by the equation where Bp(Q ) is the inverse function of Ap(R), with the initial condition

From the identity we get by differentiating whence, on account of (17.4), there follows the symmetry of dBp(Q; R ), and from it the integrability of (17.6). The uniquely determined functions Pp, and Pp are in virtue of (17.5) and (17.7) inverses one of another. Thus, we change the substratum from .9 to :7’ by effecting the transformation Pp,. To the contravariant vector A p there corresponds in I a new element A p,, for which

or

On comparing this with (17.3), we get

whence it follows, dP,(R) admitting a unique inverse, that

We see that the vector Ap, arisen from a vector Fpo(R) by 341 parallel displacement, has in the substratum I’ a representing element A’, (R) which does not depend on the point P’. To vectors at different points, parallel by teleparallelism, corresponds in 57 the same element. Thus -9 ’ is the preferred substratum for affine euclidean manifold, and g is such a manifold. If the -affine euclidean manifold is metric (and consequently also riemannian), it is called metric euclidean. For the preferred substratum 57 in such a manifold the transform of metric g(X, Y) to 9, that is g(Q, R), does not depend on the point P. The length of straight line segment P = Pi + (P2-P1)t [Otl] is then equal to

We content ourselves with these introductory developements hoping to have shown clearly enough how to transport geometric research on the basis of abstract spaces. (Received, May 18th, 1936.)